Watch the seventh video session on “Quadratic Equations” for Class Xth – concepts and solved questions
Quadratic Equations – Class X – Part 7
When we equate quadratic polynomial of the form ax2 + bx + c, to zero, we get a quadratic equation.
A quadratic equation in the variable x is an equation of the form ax2 + bx + c = 0, where a, b, c are real numbers, a ≠ 0. Any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2, is a quadratic equation. ax2 + bx + c = 0, a ≠ 0 is called the standard form of a quadratic equation.
A real number α is called a root of the quadratic equation
ax2 + bx + c = 0, a ≠ 0 if aα2 + bα + c = 0.
A quadratic polynomial can have at most two zeroes. So, any quadratic equation can have atmost two roots.
The roots of ax2 + bx + c = 0 are [-b+√(b2 – 4ac)]/2a and [(-b-√(b2 – 4ac)]/2a, if b2 – 4ac ＞= 0. If b2 – 4ac ＜ 0, the equation will have no real roots.
Exercise 4.4 Q1-5 Solved
1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
(i) 2x2 – 3x + 5 = 0
(ii) 3x2 – 4√3 x + 4 = 0
(iii) 2x2 – 6x + 3 = 0
2. Find the values of k for each of the following quadratic equations, so that they have two equal roots.
(i) 2x2 + kx + 3 = 0
(ii) kx(x – 2) + 6 = 0
3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.
4. Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
5. Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.